2022 AMC 12B Problems/Problem 4
Problem 4
For how many values of the constant will the polynomial
have two distinct integer roots?
Solution
Using Vieta's, denote the two distinct integer roots as . Then
We list out the factors of
as follows:
Then, we notice that there are exactly 4 pairings of distinct integers that multiply to
(everything excluding
).
However, we aren't done yet! and
can be negative. Since the signs of
and
must match and being negative doesn't change the number of options, we simply double the number of positive pairings to arrive at
~
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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